The Collatz conjecture: The joys of periodicity, breaking ergodicity, and mappings to infinity
The Collatz conjecture is very simple to state, and yet the challenge of proving it is still open to date. This does not mean that we cannot derive very interesting results about the conjecture. Take any natural number, 1,2,3,… If the number is even, divide it by 2; if the number is odd, multiply it by 3 and add 1. Submit the resulting number to the same procedure and so on. For instance, the number 13 (odd) gives 3*13+1=40. Next, 40 (even) is divided by 2, and so on: 20,10,5,16,8,4,2,1,4,… The conjecture is that, as in the example, the sequence ends in the ‘trivial cycle’ 4,2,1,4 for all natural numbers if the procedure is repeated a sufficient but finite number of times. If this is true, no numbers end up in a cycle other than the trivial cycle, and no numbers are mapped infinitely many times without getting into a cycle, thus going to infinity.
In this presentation I investigate the latter possibility, infinity mappings. To this end I prove that the Collatz procedure gives rise to a mapping F, the domain of which is the complete set of natural numbers partitioned as the union of two setoids, sets in which equivalence relationships are retained. All equivalent subsets in each setoid of the domain of F map to the same set of numbers (range). I show that all the subsets of both the range and the domain of F are distributed over the set of natural numbers periodically, and that this periodicity is conserved over an arbitrary large number of mappings through F (the range of F is distributed periodically over the domain). I show that while F multiplies all natural numbers by 1 on the average (ensemble average), the time average is multiplication by ¾. These combined results lead to the following paradox. On the one hand, there exist infinitely many natural numbers that follow any specified pattern of mappings through restrictions of F (which will be defined). This goes for patterns of any length, without limit. As one of the restrictions maps to higher numbers, this seems to imply that infinity mappings exist. On the other hand, after any arbitrary pattern of mappings through restrictions of F, the time average applies, which is multiplication by ¾. This seems to suggest that infinity mappings do not exist. I discuss a notion of infinity that supports the conjecture.